en>Yobot: /* Equation */Reference before punctuation using AWB (9585)

2013-11-07T16:00:49Z

Equation: Reference before punctuation using AWB (9585)

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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Andrew~~ and ~~my spouse doesn~~'~~t like~~ it ~~at all~~. ~~Mississippi~~ is where ~~his house~~ is. He is an ~~information officer~~. The ~~favorite pastime~~ for ~~him~~ and ~~his kids~~ is to ~~perform lacross and he would never give it up~~.<br><br>~~my web page~~ [http://~~www~~.~~010~~-~~5260~~-~~5333~~.~~com/index~~.~~php?document_srl~~=~~1880~~&~~mid~~=~~board_ALMP66 real psychic~~]		[[File:ExtremeGaussian.png\|thumb\|The graph of the function in gray is <math>e^{-\pi z^{2}}</math>, the Gaussian restricted to the real axis. The Gaussian does not have exponential type, but the functions in red and blue are one sided approximations that have exponential type <math>2\pi</math>.]]

			In [[complex analysis]], a branch of [[mathematics]], a [[holomorphic function]] is said to be of '''exponential type C''' if its growth is bounded by the [[exponential function]] ''e''<sup>''C\|z\|''</sup> for some constant ''C'' as \|''z''\|→∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as [[Borel summation]], or, for example, to apply the [[Mellin transform]], or to perform approximations using the [[Euler-MacLaurin formula]]. The general case is handled by [[Nachbin's theorem]], which defines the analogous notion of '''Ψ-type''' for a general function ''Ψ(z)'' as opposed to ''e''<sup>''z''</sup>.

			==Basic idea==

			A function ''f''(''z'') defined on the [[complex plane]] is said to be of exponential type if there exist constants ''M'' and τ such that

			:<math>\|f(re^{i\theta})\|\le Me^{\tau r}</math>

			in the limit of <math>r\to\infty</math>. Here, the [[complex variable]] ''z'' was written as <math>z=re^{i\theta}</math> to emphasize that the limit must hold in all directions θ. Letting τ stand for the [[infimum]] of all such τ, one then says that the function ''f'' is of ''exponential type τ''.

			For example, let <math>f(z)=\sin(\pi z)</math>. Then one says that <math>\sin(\pi z)</math> is of exponential type π, since π is the smallest number that bounds the growth of <math>\sin(\pi z)</math> along the imaginary axis. So, for this example, [[Carlson's theorem]] cannot apply, as it requires functions of exponential type less than π. Similarly, the [[Euler-MacLaurin formula]] cannot be applied either, as it, too, expresses an theorem ultimately anchored in the theory of [[finite difference]]s.

			==Formal definition==
			A [[holomorphic function]] function <math>F(z)</math> is said to be of '''exponential type''' <math>\sigma>0</math> if for every <math>\varepsilon>0</math> there exists a constant <math>A_\varepsilon </math> such that

			:<math>\|F(z)\|\leq A_\varepsilon e^{(\sigma+\varepsilon)\|z\|}</math>

			for <math>\|z\|\to\infty</math> where <math>z\in\mathbb{C}</math>.
			We say <math>F(z)</math> is of exponential type if <math>F(z)</math> is of exponential type <math>\sigma</math> for some <math>\sigma>0</math>. The number

			:<math>\tau(F)=\sigma=\displaystyle\limsup_{\|z\|\rightarrow\infty}\|z\|^{-1}\log\|F(z)\|</math>

			is the exponential type of <math>F(z)</math>.

			==Exponential type with respect to a symmetric convex body==

			{{harvtxt\|Stein\|1957}} has given a generalization of exponential type for [[entire function]]s of [[several complex variables]].
			Suppose <math>K</math> is a [[convex set\|convex]], [[compact element\|compact]], and [[symmetric]] subset of <math>\mathbb{R}^n</math>. It is known that for every such <math>K</math> there is an associated [[norm (mathematics)\|norm]] <math>\\|\cdot\\|_K</math> with the property that
			:<math>K=\{x\in\mathbb{R}^n : \\|x\\|_K \leq1\}</math>.
			In other words, <math>K</math> is the unit ball in <math>\mathbb{R}^{n}</math> with respect to <math>\\|\cdot\\|_K</math>.
			The set
			:<math>K^{*}=\{y\in\mathbb{R}^{n}:x\cdot y \leq 1 \text{ for all }x\in{K}\}</math>
			is called the [[polar set]] and is also a [[convex set\|convex]], [[compact element\|compact]], and [[symmetric]] subset of <math>\mathbb{R}^n</math>. Furthermore, we can write
			:<math>\\|x\\|_K = \displaystyle\sup_{y\in K^{*}}\|x\cdot y\|</math>.

			We extend <math>\\|\cdot\\|_K</math> from <math>\mathbb{R}^n</math> to <math>\mathbb{C}^n</math> by

			:<math>\\|z\\|_K = \displaystyle\sup_{y\in K^{*}}\|z\cdot y\|.</math>

			An entire function <math>F(z)</math> of <math>n</math>-complex variables is said to be of exponential type with respect to <math>K</math> if for every <math>\varepsilon>0</math> there exists a constant <math>A_\varepsilon</math> such that
			:<math>\|F(z)\|<A_\varepsilon e^{2\pi(1+\varepsilon)\\|z\\|_K}</math>
			for all <math>z\in\mathbb{C}^{n}</math>.

			==Fréchet space==
			Collections of functions of exponential type <math>\tau</math> can form a [[complete space\|complete]] [[uniform space]], namely a [[Fréchet space]], by the [[topological space\|topology]] induced by the countable family of [[norm (mathematics)\|norm]]s

			:<math> \\|f\\|_{n} = \sup_{z \in \mathbb{C}} \exp \left[-\left(\tau + \frac{1}{n}\right)\|z\|\right]\|f(z)\| </math>

			==See also==
			*[[Paley–Wiener theorem]]
			*[[Paley–Wiener space]]

			== References ==
			<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
			{{Reflist}}

			* {{citation
			\| last = Stein
			\| first = E.M.
			\| author-link = Elias M. Stein
			\| title = Functions of exponential type
			\| journal = Ann. of Math. (2)
			\| volume = 65
			\| year = 1957
			\| pages = 582–592
			\| mr = 0085342
			\| jstor = 1970066
			}}

			[[Category:Complex analysis]]

213.243.132.247: cleanup: extra tag removed

2012-06-11T15:02:44Z

cleanup: extra </math> tag removed

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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Mississippi is where his house is. He is an information officer. The favorite pastime for him and his kids is to perform lacross and he would never give it up.<br><br>my web page [http://www.010-5260-5333.com/index.php?document_srl=1880&mid=board_ALMP66 real psychic]

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en>Yobot: /* Equation */Reference before punctuation using AWB (9585)

213.243.132.247: cleanup: extra tag removed